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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.049]   [H-I: 102]   [3 followers]  Follow
    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [3039 journals]
  • Low-dimensional reduced-order models for statistical response and
           uncertainty quantification: Barotropic turbulence with topography
    • Authors: Di Qi; Andrew J. Majda
      Pages: 7 - 27
      Abstract: Publication date: 15 March 2017
      Source:Physica D: Nonlinear Phenomena, Volume 343
      Author(s): Di Qi, Andrew J. Majda
      A low-dimensional reduced-order statistical closure model is developed for quantifying the uncertainty to changes in forcing in a barotropic turbulent system with topography involving interactions between small-scale motions and a large-scale mean flow. Imperfect model sensitivity is improved through a recent mathematical strategy for calibrating model errors in a training phase, where information theory and linear statistical response theory are combined in a systematic fashion to achieve the optimal model parameters. Statistical theories about a Gaussian invariant measure and the exact statistical energy equations are also developed for the truncated barotropic equations that can be used to improve the imperfect model prediction skill. A stringent paradigm model of 57 degrees of freedom is used to display the feasibility of the reduced-order methods. This simple model creates large-scale zonal mean flow shifting directions from westward to eastward jets with an abrupt change in amplitude when perturbations are applied, and prototype blocked and unblocked patterns can be generated in this simple model similar to the real natural system. Principal statistical responses in mean and variance can be captured by the reduced-order models with desirable accuracy and efficiency with only 3 resolved modes. An even more challenging regime with non-Gaussian equilibrium statistics using the fluctuation equations is also tested in the reduced-order models with accurate prediction using the first 5 resolved modes. These reduced-order models also show potential for uncertainty quantification and prediction in more complex realistic geophysical turbulent dynamical systems.

      PubDate: 2017-02-09T06:01:16Z
      DOI: 10.1016/j.physd.2016.11.006
      Issue No: Vol. 343 (2017)
       
  • Isolating blocks as computational tools in the circular restricted
           three-body problem
    • Authors: Rodney L. Anderson; Robert W. Easton; Martin W. Lo
      Pages: 38 - 50
      Abstract: Publication date: 15 March 2017
      Source:Physica D: Nonlinear Phenomena, Volume 343
      Author(s): Rodney L. Anderson, Robert W. Easton, Martin W. Lo
      Isolating blocks may be used as computational tools to search for the invariant manifolds of orbits and hyperbolic invariant sets associated with libration points while also giving additional insight into the dynamics of the flow in these regions. We use isolating blocks to investigate the dynamics of objects entering the Earth–Moon system in the circular restricted three-body problem with energies close to the energy of the L 2 libration point. Specifically, the stable and unstable manifolds of Lyapunov orbits and the hyperbolic invariant set around the libration points are obtained by numerically computing the way orbits exit from an isolating block in combination with a bisection method. Invariant spheres of solutions in the spatial problem may then be located using the resulting manifolds.

      PubDate: 2017-02-09T06:01:16Z
      DOI: 10.1016/j.physd.2016.10.004
      Issue No: Vol. 343 (2017)
       
  • Finite-time thin film rupture driven by modified evaporative loss
    • Authors: Hangjie Ji; Thomas P. Witelski
      Pages: 1 - 15
      Abstract: Publication date: 1 March 2017
      Source:Physica D: Nonlinear Phenomena, Volume 342
      Author(s): Hangjie Ji, Thomas P. Witelski
      Rupture is a nonlinear instability resulting in a finite-time singularity as a film layer approaches zero thickness at a point. We study the dynamics of rupture in a generalized mathematical model of thin films of viscous fluids with modified evaporative effects. The governing lubrication model is a fourth-order nonlinear parabolic partial differential equation with a non-conservative loss term. Several different types of finite-time singularities are observed due to balances between conservative and non-conservative terms. Non-self-similar behavior and two classes of self-similar rupture solutions are analyzed and validated against high resolution PDE simulations.

      PubDate: 2017-01-14T04:47:15Z
      DOI: 10.1016/j.physd.2016.10.002
      Issue No: Vol. 342 (2017)
       
  • Stability on time-dependent domains: convective and dilution effects
    • Authors: R. Krechetnikov; E. Knobloch
      Pages: 16 - 23
      Abstract: Publication date: 1 March 2017
      Source:Physica D: Nonlinear Phenomena, Volume 342
      Author(s): R. Krechetnikov, E. Knobloch
      We explore near-critical behavior of spatially extended systems on time-dependent spatial domains with convective and dilution effects due to domain flow. As a paradigm, we use the Swift–Hohenberg equation, which is the simplest nonlinear model with a non-zero critical wavenumber, to study dynamic pattern formation on time-dependent domains. A universal amplitude equation governing weakly nonlinear evolution of patterns on time-dependent domains is derived and proves to be a generalization of the standard Ginzburg–Landau equation. Its key solutions identified here demonstrate a substantial variety–spatially periodic states with a time-dependent wavenumber, steady spatially non-periodic states, and pulse-train solutions–in contrast to extended systems on time-fixed domains. The effects of domain flow, such as bifurcation delay due to domain growth and destabilization due to oscillatory domain flow, on the Eckhaus instability responsible for phase slips in spatially periodic states are analyzed with the help of both local and global stability analyses. A nonlinear phase equation describing the approach to a phase-slip event is derived. Detailed analysis of a phase slip using multiple time scale methods demonstrates different mechanisms governing the wavelength changing process at different stages.

      PubDate: 2017-01-14T04:47:15Z
      DOI: 10.1016/j.physd.2016.10.003
      Issue No: Vol. 342 (2017)
       
  • Assimilating Eulerian and Lagrangian data in traffic-flow models
    • Authors: Chao Xia; Courtney Cochrane; Joseph DeGuire; Gaoyang Fan; Emma Holmes; Melissa McGuirl; Patrick Murphy; Jenna Palmer; Paul Carter; Laura Slivinski; Björn Sandstede
      Abstract: Publication date: Available online 21 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Chao Xia, Courtney Cochrane, Joseph DeGuire, Gaoyang Fan, Emma Holmes, Melissa McGuirl, Patrick Murphy, Jenna Palmer, Paul Carter, Laura Slivinski, Björn Sandstede
      Data assimilation of traffic flow remains a challenging problem. One difficulty is that data come from different sources ranging from stationary sensors and camera data to GPS and cell phone data from moving cars. Sensors and cameras give information about traffic density, while GPS data provide information about the positions and velocities of individual cars. Previous methods for assimilating Lagrangian data collected from individual cars relied on specific properties of the underlying computational model or its reformulation in Lagrangian coordinates. These approaches make it hard to assimilate both Eulerian density and Lagrangian positional data simultaneously. In this paper, we propose an alternative approach that allows us to assimilate both Eulerian and Lagrangian data. We show that the proposed algorithm is accurate and works well in different traffic scenarios and regardless of whether ensemble Kalman or particle filters are used. We also show that the algorithm is capable of estimating parameters and assimilating real traffic observations and synthetic observations obtained from microscopic models.

      PubDate: 2017-02-21T09:49:54Z
      DOI: 10.1016/j.physd.2017.02.004
       
  • Complex predator invasion waves in a Holling-Tanner model with nonlocal
           prey interaction
    • Authors: A. Bayliss; V.A. Volpert
      Abstract: Publication date: Available online 20 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): A. Bayliss, V.A. Volpert
      We consider predator invasions for the nonlocal Holling-Tanner model. Predators are introduced in a small region adjacent to an extensive predator-free region. In its simplest form an invasion front propagates into the predator-free region with a predator-prey coexistence state displacing the predator-free state. However, patterns may form in the wake of the invasion front due to instability of the coexistence state. The coexistence state can be subject to either oscillatory or cellular instability, depending on parameters. Furthermore, the oscillatory instability can be either at zero wave number or finite wave number. In addition, the (unstable) predator-free state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large. We perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors. These regions are separated by relatively narrow transition regions that we refer to as fronts. We also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations.

      PubDate: 2017-02-21T09:49:54Z
      DOI: 10.1016/j.physd.2017.02.003
       
  • Elementary solutions for a model Boltzmann equation in one dimension and
           the connection to grossly determined solutions
    • Authors: Thomas E. Carty
      Abstract: Publication date: Available online 20 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Thomas E. Carty
      The Fourier-transformed version of the time dependent slip-flow model Boltzmann equation associated with the linearized BGK model is solved in order to determine the solution’s asymptotics. The ultimate goal of this paper is to demonstrate that there exists a robust set of solutions to this model Boltzmann equation that possess a special property that was conjectured by Truesdell and Muncaster: that solutions decay to a subclass of the solution set uniquely determined by the initial mass density of the gas called the grossly determined solutions. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case’s method of elementary solutions, we find analytic time-dependent solutions to the model Boltzmann equation for initial data with a specialized compact support condition under the Fourier transform. In doing so, we show that the spectrum separates the solutions into two distinct parts: one that behaves as a set of transient solutions and the other limiting to a stable subclass of solutions. Thus, we demonstrate that for gas flows with this specialized initial density condition, in time all gas flows for the one dimensional model Boltzmann equation act as grossly determined solutions.

      PubDate: 2017-02-21T09:49:54Z
      DOI: 10.1016/j.physd.2017.02.008
       
  • Bright and dark solitons in the unidirectional long wave limit for the
           energy transfer on anharmonic crystal lattices
    • Authors: Luis A. Cisneros-Ake; José F. Solano Peláez
      Abstract: Publication date: Available online 9 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Luis A. Cisneros-Ake, José F. Solano Peláez
      The problem of energy transportation along a cubic anharmonic crystal lattice, in the unidirectional long wave limit, is considered. A detailed process, in the discrete lattice equations, shows that unidirectional stable propagating waves for the continuum limit produces a coupled system between a nonlinear Schrödinger (NLS) equation and the Korteweg-deVries (KdV) equation. The traveling wave formalism provides a diversity of exact solutions ranging from the classical Davydov’s soliton (subsonic and supersonic) of the first and second kind to a class consisting in the coupling between the KdV soliton and dark solitons containing the typical ones (similar to the dark-gray soliton in the standard defocusing NLS) and a new kind in the form of a two-hump dark soliton. This family of exact solutions are numerically tested, by means of the pseudo spectral method, in our NLS-KdV system.

      PubDate: 2017-02-15T09:34:42Z
      DOI: 10.1016/j.physd.2017.02.001
       
  • The Lyapunov-Krasovskii theorem and a sufficient criterion for local
           stability of isochronal synchronization in networks of delay-coupled
           oscillators
    • Authors: J.M.V. Grzybowski; E.E.N. Macau; T. Yoneyama
      Abstract: Publication date: Available online 9 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): J.M.V. Grzybowski, E.E.N. Macau, T. Yoneyama
      This paper presents a self-contained framework for the stability assessment of isochronal synchronization in networks of chaotic and limit-cycle oscillators. The results were based on the Lyapunov-Krasovskii theorem and they establish a sufficient condition for local synchronization stability of as a function of the system and network parameters. With this in mind, a network of mutually delay-coupled oscillators subject to direct self-coupling is considered and then the resulting error equations are block-diagonalized for the purpose of studying their stability. These error equations are evaluated by means of analytical stability results derived from the Lyapunov-Krasovskii theorem. The proposed approach is shown to be a feasible option for the investigation of local stability of isochronal synchronization for a variety of oscillators coupled through linear functions of the state variables under a given undirected graph structure. This ultimately permits the systematic identification of stability regions within the high-dimensionality of the network parameter space. Examples of applications of the results to a number of networks of delay-coupled chaotic and limit-cycle oscillators are provided, such as Lorenz, Rössler, Cubic Chua’s circuit, Van der Pol oscillator and the Hindmarsh-Rose neuron.

      PubDate: 2017-02-15T09:34:42Z
      DOI: 10.1016/j.physd.2017.01.005
       
  • Wave fronts and cascades of soliton interactions in the periodic two
           dimensional Volterra system
    • Authors: Rhys Bury; Alexander V. Mikhailov; Jing Ping Wang
      Abstract: Publication date: Available online 6 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Rhys Bury, Alexander V. Mikhailov, Jing Ping Wang
      In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N . We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmannians Gr R ( k , N ) and Gr C ( k , N ) .

      PubDate: 2017-02-09T06:01:16Z
      DOI: 10.1016/j.physd.2017.01.003
       
  • The stability spectrum for elliptic solutions to the focusing NLS equation
    • Authors: Bernard Deconinck; Benjamin L. Segal
      Abstract: Publication date: Available online 23 January 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Bernard Deconinck, Benjamin L. Segal
      We present an analysis of the stability spectrum of all stationary elliptic-type solutions to the focusing Nonlinear Schrödinger equation (NLS). An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about the spectrum are derived. Specifically, the solution parameter space is shown to be split into four regions of distinct qualitative behavior of the spectrum. Additional results on the stability of solutions with respect to perturbations of an integer multiple of the period are given, as well as a procedure for approximating the greatest real part of the spectrum.

      PubDate: 2017-01-28T05:15:28Z
      DOI: 10.1016/j.physd.2017.01.004
       
  • Wave turbulence theory of elastic plates
    • Authors: Gustavo Düring; Christophe Josserand; Sergio Rica
      Abstract: Publication date: Available online 16 January 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Gustavo Düring, Christophe Josserand, Sergio Rica
      This article presents the complete study of the long-time evolution of random waves of a vibrating thin elastic plate in the limit of small plate deformation so that modes of oscillations interact weakly. According to the wave turbulence theory a nonlinear wave system evolves in longtime creating a slow redistribution of the spectral energy from one mode to another. We derive step by step, following the method of cumulants expansion and multiscale asymptotic perturbations, the kinetic equation for the second order cumulants as well as the second and fourth order renormalization of the dispersion relation of the waves. We characterize the non-equilibrium evolution to an equilibrium wave spectrum, which happens to be the well known Rayleigh-Jeans distribution. Moreover we show the existence of an energy cascade, often called the Kolmogorov-Zakharov spectrum, which happens to be not simply a power law, but a logarithmic correction to the Rayleigh-Jeans distribution. We perform numerical simulations confirming these scenarii, namely the equilibrium relaxation for closed systems and the existence of an energy cascade wave spectrum. Both show a good agreement between theoretical predictions and numerics. We show also some other relevant features of vibrating elastic plates, such as the existence of a self-similar wave action inverse cascade which happens to blow-up in finite time. We discuss the mechanism of the wave breakdown phenomena in elastic plates as well as the limit of strong turbulence which arises as the thickness of the plate vanishes. Finally, we discuss the role of dissipation and the connection with experiments, and the generalization of the wave turbulence theory to elastic shells.

      PubDate: 2017-01-21T05:03:49Z
      DOI: 10.1016/j.physd.2017.01.002
       
  • Full analysis of small hypercycles with short-circuits in prebiotic
           evolution
    • Authors: Josep Sardanyés; J. Tomás Lázaro; Toni Guillamon; Ernest Fontich
      Abstract: Publication date: Available online 5 January 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Josep Sardanyés, J. Tomás Lázaro, Toni Guillamon, Ernest Fontich
      It is known that hypercycles are sensitive to the so-called parasites and short-circuits. While the impact of parasites has been widely investigated for well-mixed and spatial hypercycles, the effect of short-circuits in hypercycles remains poorly understood. In this article we analyze the mean field and spatial dynamics of two small, asymmetric hypercycles with short-circuits. Specifically, we analyze a two-member hypercycle where one of the species contains an autocatalytic loop, as the simplest hypercycle with a short-circuit. Then, we extend this system by adding another species that closes a three-member hypercycle while keeping the autocatalytic short-circuit and the two-member cycle. The mean field model allows us to discard the presence of stable or unstable periodic orbits for both systems. We characterize the bifurcations and transitions involved in the dominance of the short-circuits i.e., in the reduction of the hypercycles’ size. The spatial simulations reveal a random-like and mixed distribution of the replicators in the all-species coexistence, ruling out the presence of large-scale spatial patterns such as spirals or spots typical of larger, oscillating hypercycles. A MonteCarlo sampling of the parameter space for the well-mixed and the spatial models reveals that the probability of finding stable hypercycles with short-circuits drastically diminishes from the two-member to the three-member system, especially at growing degradation rates of the replicators. These findings pose a big constrain in the increase of hypercycle’s size and complexity under the presence of inner cycles, suggesting the importance of a rapid growth of hypercycles able to generate spatial structures (e.g., rotating spirals) prior to the emergence of inner cycles. Our results can also be useful for the future design and implementation of synthetic cooperative systems containing catalytic short-circuits.

      PubDate: 2017-01-14T04:47:15Z
      DOI: 10.1016/j.physd.2016.12.004
       
  • A numerical estimate of the regularity of a family of Strange
           Non–Chaotic Attractors
    • Authors: Lluís Alsedà i Soler; Josep Maria Mondelo González; David Romero i Sànchez
      Abstract: Publication date: Available online 5 January 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Lluís Alsedà i Soler, Josep Maria Mondelo González, David Romero i Sànchez
      We estimate numerically the regularities of a family of Strange Non–Chaotic Attractors related with one of the models studied in Grebogi et al. (1984) (see also Keller, 1996). To estimate these regularities we use wavelet analysis in the spirit of Llave and Petrov (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.

      PubDate: 2017-01-14T04:47:15Z
      DOI: 10.1016/j.physd.2016.12.006
       
  • Reduced-space Gaussian Process Regression for data-driven probabilistic
           forecast of chaotic dynamical systems
    • Authors: Zhong Yi Wan; Themistoklis P. Sapsis
      Abstract: Publication date: Available online 5 January 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Zhong Yi Wan, Themistoklis P. Sapsis
      We formulate a reduced-order strategy for efficiently forecasting complex high-dimensional dynamical systems entirely based on data streams. The first step of our method involves reconstructing the dynamics in a reduced-order subspace of choice using Gaussian Process Regression (GPR). GPR simultaneously allows for reconstruction of the vector field and more importantly, estimation of local uncertainty. The latter is due to (i) local interpolation error and (ii) truncation of the high-dimensional phase space. This uncertainty component can be analytically quantified in terms of the GPR hyperparameters. In the second step we formulate stochastic models that explicitly take into account the reconstructed dynamics and their uncertainty. For regions of the attractor which are not sufficiently sampled for our GPR framework to be effective, an adaptive blended scheme is formulated to enforce correct statistical steady state properties, matching those of the real data. We examine the effectiveness of the proposed method to complex systems including the Lorenz 96, the Kuramoto-Sivashinsky, as well as a prototype climate model. We also study the performance of the proposed approach as the intrinsic dimensionality of the system attractor increases in highly turbulent regimes.

      PubDate: 2017-01-14T04:47:15Z
      DOI: 10.1016/j.physd.2016.12.005
       
  • Characterizing complex networks through statistics of Möbius
           transformations
    • Authors: Vladimir Jaćimović; Aladin Crnkić
      Abstract: Publication date: Available online 31 December 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Vladimir Jaćimović, Aladin Crnkić
      It is well-known now that dynamics of large populations of globally (all-to-all) coupled oscillators can be reduced to low-dimensional submanifolds (WS transformation and OA ansatz). Marvel et al. (2009) described an intriguing algebraic structure standing behind this reduction: oscillators evolve by the action of the group of Möbius transformations. Of course, dynamics in complex networks of coupled oscillators is highly complex and not reducible. Still, closer look unveils that even in complex networks some (possibly overlapping) groups of oscillators evolve by Möbius transformations. In this paper we study properties of the network by identifying Möbius transformations in the dynamics of oscillators. This enables us to introduce some new (statistical) concepts that characterize the network. In particular, the notion of coherence of the network (or subnetwork) is proposed. This conceptual approach is meaningful for the broad class of networks, including those with time-delayed, noisy or mixed interactions. In this paper several simple (random) graphs are studied illustrating the meaning of the concepts introduced in the paper.

      PubDate: 2017-01-05T04:02:55Z
      DOI: 10.1016/j.physd.2016.12.007
       
  • Classical quasi-steady state reduction—A mathematical
           characterization
    • Authors: Alexandra Goeke; Sebastian Walcher; Eva Zerz
      Abstract: Publication date: Available online 30 December 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexandra Goeke, Sebastian Walcher, Eva Zerz
      We discuss parameter dependent polynomial ordinary differential equations that model chemical reaction networks. By classical quasi-steady state (QSS) reduction we understand the following familiar (heuristically motivated) mathematical procedure: Set the rate of change for certain (a priori chosen) variables equal to zero and use the resulting algebraic equations to obtain a system of smaller dimension for the remaining variables. This procedure will generally be valid only for certain parameter ranges. We start by showing that the reduction is accurate if and only if the corresponding parameter is what we call a QSS parameter value, and that the reduction is approximately accurate if and only if the corresponding parameter is close to a QSS parameter value. The QSS parameter values can be characterized by polynomial equations and inequations, hence parameter ranges for which QSS reduction is valid are accessible in an algorithmic manner. A defining characteristic of a QSS parameter value is that the algebraic variety defined by the QSS relations is invariant for the differential equation. A closer investigation of the associated systems shows the existence of further invariant sets; here singular perturbations enter the picture in a natural manner. We compare QSS reduction and singular perturbation reduction, and show that, while they do not agree in general, they do, up to lowest order in a small parameter, for a quite large and relevant class of examples. This observation, in turn, allows the computation of QSS reductions even in cases where an explicit resolution of the polynomial equations is not possible.

      PubDate: 2017-01-05T04:02:55Z
      DOI: 10.1016/j.physd.2016.12.002
       
  • Cascades of alternating pitchfork and flip bifurcations in H-bridge
           inverters
    • Authors: Viktor Avrutin; Zhanybai T. Zhusubaliyev; Erik Mosekilde
      Abstract: Publication date: Available online 28 December 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Viktor Avrutin, Zhanybai T. Zhusubaliyev, Erik Mosekilde
      Power electronic DC/AC converters (inverters) play an important role in modern power engineering. These systems are also of considerable theoretical interest because their dynamics is influenced by the presence of two vastly different forcing frequencies. As a consequence, inverter systems may be modeled in terms of piecewise smooth maps with an extremely high number of switching manifolds. We have recently shown that models of this type can demonstrate a complicated bifurcation structure associated with the occurrence of border collisions. Considering the example of a PWM H-bridge single-phase inverter, the present paper discusses a number of unusual phenomena that can occur in piecewise smooth maps with a very large number of switching manifolds. We show in particular how smooth (pitchfork and flip) bifurcations may form a macroscopic pattern that stretches across the overall bifurcation structure. We explain the observed bifurcation phenomena, show under which conditions they occur, and describe them quantitatively by means of an analytic approximation.

      PubDate: 2017-01-05T04:02:55Z
      DOI: 10.1016/j.physd.2016.12.008
       
  • Spiral disk packings
    • Authors: Yoshikazu Yamagishi; Takamichi Sushida
      Abstract: Publication date: Available online 26 December 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yoshikazu Yamagishi, Takamichi Sushida
      It is shown that van Iterson’s metric for disk packings, proposed in 1907 in the study of a centric model of spiral phyllotaxis, defines a bounded distance function in the plane. This metric is also related to the bifurcation of Voronoi tilings for logarithmic spiral lattices, through the continued fraction expansion of the divergence angle. The phase diagrams of disk packings and Voronoi tilings for logarithmic spirals are dual graphs to each other. This gives a rigorous proof that van Iterson’s diagram in the centric model is connected and simply connected. It is a nonlinear analog of the duality between the phase diagrams for disk packings and Voronoi tilings on the linear lattices, having the modular group symmetry.

      PubDate: 2017-01-05T04:02:55Z
      DOI: 10.1016/j.physd.2016.12.003
       
  • Complicated quasiperiodic oscillations and chaos from driven
           piecewise-constant circuit: Chenciner bubbles do not necessarily occur via
           simple phase-locking
    • Authors: Tri Quoc Truong; Tadashi Tsubone; Munehisa Sekikawa; Naohiko Inaba
      Pages: 1 - 9
      Abstract: Publication date: 15 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 341
      Author(s): Tri Quoc Truong, Tadashi Tsubone, Munehisa Sekikawa, Naohiko Inaba
      We analyze the complex quasiperiodic oscillations and chaos generated by two coupled piecewise-constant hysteresis oscillators driven by a rectangular wave force. Oscillations generate Arnol’d resonance webs wherein lower dimensional resonance tongues extend such as that of a web in numerous directions. We provide the fundamental tools for bifurcation analysis of nonautonomous piecewise-constant oscillators. To optimize the outstanding simplicity of piecewise-constant circuits, we formulate a generalized procedure for calculating the Lyapunov exponents in nonautonomous piecewise-constant dynamics. The Lyapunov exponents in these dynamics can be calculated with a precision approximately similar to that of maps. We observe two-parameter Lyapunov diagrams near the fundamental resonance region called Chenciner bubbles, wherein the oscillation frequencies of the two oscillators and the force are synchronized with a ratio of 1:1:1. Inevitably, the hysteresis considerably distorts the Chenciner bubbles. This result suggests that the Chenciner bubbles do not necessarily occur due to simple phase-locking of two-dimensional tori that can be explained by homeomorphism on the circle. Furthermore, we observe the Farey sequence in the experimental measurements.

      PubDate: 2016-12-26T03:35:57Z
      DOI: 10.1016/j.physd.2016.09.008
      Issue No: Vol. 341 (2016)
       
  • Gaussian noise and the two-network frustrated Kuramoto model
    • Authors: Andrew B. Holder; Mathew L. Zuparic; Alexander C. Kalloniatis
      Pages: 10 - 32
      Abstract: Publication date: 15 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 341
      Author(s): Andrew B. Holder, Mathew L. Zuparic, Alexander C. Kalloniatis
      We examine analytically and numerically a variant of the stochastic Kuramoto model for phase oscillators coupled on a general network. Two populations of phased oscillators are considered, labelled ‘Blue’ and ‘Red’, each with their respective networks, internal and external couplings, natural frequencies, and frustration parameters in the dynamical interactions of the phases. We disentangle the different ways that additive Gaussian noise may influence the dynamics by applying it separately on zero modes or normal modes corresponding to a Laplacian decomposition for the sub-graphs for Blue and Red. Under the linearisation ansatz that the oscillators of each respective network remain relatively phase-synchronised centroids or clusters, we are able to obtain simple closed-form expressions using the Fokker–Planck approach for the dynamics of the average angle of the two centroids. In some cases, this leads to subtle effects of metastability that we may analytically describe using the theory of ratchet potentials. These considerations are extended to a regime where one of the populations has fragmented in two. The analytic expressions we derive largely predict the dynamics of the non-linear system seen in numerical simulation. In particular, we find that noise acting on a more tightly coupled population allows for improved synchronisation of the other population where deterministically it is fragmented.

      PubDate: 2016-12-26T03:35:57Z
      DOI: 10.1016/j.physd.2016.09.009
      Issue No: Vol. 341 (2016)
       
  • Microorganism billiards
    • Authors: Saverio E. Spagnolie; Colin Wahl; Joseph Lukasik; Jean-Luc Thiffeault
      Pages: 33 - 44
      Abstract: Publication date: 15 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 341
      Author(s): Saverio E. Spagnolie, Colin Wahl, Joseph Lukasik, Jean-Luc Thiffeault
      Recent experiments and numerical simulations have shown that certain types of microorganisms “reflect” off of a flat surface at a critical angle of departure, independent of the angle of incidence. The nature of the reflection may be active (cell and flagellar contact with the surface) or passive (hydrodynamic) interactions. We explore the billiard-like motion of a body with this empirical reflection law inside a regular polygon and show that the dynamics can settle on a stable periodic orbit or can be chaotic, depending on the swimmer’s departure angle and the domain geometry. The dynamics are often found to be robust to the introduction of weak random fluctuations. The Lyapunov exponent of swimmer trajectories can be positive or negative, can have extremal values, and can have discontinuities depending on the degree of the polygon. A passive sorting device is proposed that traps swimmers of different departure angles into separate bins. We also study the external problem of a microorganism swimming in a patterned environment of square obstacles, where the departure angle dictates the possibility of trapping or diffusive trajectories.

      PubDate: 2016-12-26T03:35:57Z
      DOI: 10.1016/j.physd.2016.09.010
      Issue No: Vol. 341 (2016)
       
  • A theory of synchrony for active compartments with delays coupled through
           bulk diffusion
    • Authors: Bin Xu; Paul C. Bressloff
      Pages: 45 - 59
      Abstract: Publication date: 15 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 341
      Author(s): Bin Xu, Paul C. Bressloff
      We extend recent work on the analysis of synchronization in a pair of biochemical oscillators coupled by linear bulk diffusion, in order to explore the effects of discrete delays. More specifically, we consider two well-mixed, identical compartments located at either end of a bounded, one-dimensional domain. The compartments can exchange signaling molecules with the bulk domain, within which the signaling molecules undergo diffusion. The concentration of signaling molecules in each compartment is modeled by a delay differential equation (DDE), while the concentration in the bulk medium is modeled by a partial differential equation (PDE) for diffusion. Coupling in the resulting PDE–DDE system is via flux terms at the boundaries. Using linear stability analysis, numerical simulations and bifurcation analysis, we investigate the effect of diffusion on the onset of a supercritical Hopf bifurcation. The direction of the Hopf bifurcation is determined by numerical simulations and a winding number argument. Near a Hopf bifurcation point, we find that there are oscillations with two possible modes: in-phase and anti-phase. Moreover, the critical delay for oscillations to occur increases with the diffusion coefficient. Our numerical results suggest that the selection of the in-phase or anti-phase oscillation is sensitive to the diffusion coefficient, time delay and coupling strength. For slow diffusion and weak coupling both modes can coexist, while for fast diffusion and strong coupling, only one of the modes is dominant, depending on the explicit choice of DDE.

      PubDate: 2016-12-26T03:35:57Z
      DOI: 10.1016/j.physd.2016.10.001
      Issue No: Vol. 341 (2016)
       
  • A theory of synchrony by coupling through a diffusive chemical signal
    • Authors: Jia Gou; Wei-Yin Chiang; Pik-Yin Lai; Michael J. Ward; Yue-Xian Li
      Pages: 1 - 17
      Abstract: Publication date: 15 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 339
      Author(s): Jia Gou, Wei-Yin Chiang, Pik-Yin Lai, Michael J. Ward, Yue-Xian Li
      We formulate and analyze oscillatory dynamics associated with a model of dynamically active, but spatially segregated, compartments that are coupled through a chemical signal that diffuses in the bulk medium between the compartments. The coupling between each compartment and the bulk is due to both feedback terms to the compartmental dynamics and flux boundary conditions at the interface between the compartment and the bulk. Our coupled model consists of dynamically active compartments located at the two ends of a 1-D bulk region of spatial extent 2 L . The dynamics in the two compartments is modeled by Sel’kov kinetics, and the signaling molecule between the two-compartments is assumed to undergo both diffusion, with diffusivity D , and first-order, linear, bulk degradation. For the resulting PDE–ODE system, we construct a symmetric steady-state solution and analyze the stability of this solution to either in-phase synchronous or anti-phase synchronous perturbations about the midline x = L . The conditions for the onset of oscillatory dynamics, as obtained from a linearization of the steady-state solution, are studied using a winding number approach. Global branches of either in-phase or anti-phase periodic solutions, and their associated stability properties, are determined numerically. For the case of a linear coupling between the compartments and the bulk, with coupling strength β , a phase diagram, in the parameter space D versus β is constructed that shows the existence of a rather wide parameter regime where stable in-phase synchronized oscillations can occur between the two compartments. By using a Floquet-based approach, this analysis with linear coupling is then extended to determine Hopf bifurcation thresholds for a periodic chain of evenly-spaced dynamically active units. Finally, we consider one particular case of a nonlinear coupling between two active compartments and the bulk. It is shown that stable in-phase and anti-phase synchronous oscillations also occur in certain parameter regimes, but as isolated solution branches that are disconnected from the steady-state solution branch.

      PubDate: 2016-11-27T16:42:30Z
      DOI: 10.1016/j.physd.2016.08.004
      Issue No: Vol. 339 (2016)
       
  • Global dynamics for steep nonlinearities in two dimensions
    • Authors: Tomáš Gedeon; Shaun Harker; Hiroshi Kokubu; Konstantin Mischaikow; Hiroe Oka
      Pages: 18 - 38
      Abstract: Publication date: 15 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 339
      Author(s): Tomáš Gedeon, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka
      This paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. We study switching models of regulatory networks. To each switching network we associate a Morse graph, a computable object that describes a Morse decomposition of the dynamics. In this paper we show that all smooth perturbations of the switching system share the same Morse graph and we compute explicit bounds on the size of the allowable perturbation. This shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities.

      PubDate: 2016-11-27T16:42:30Z
      DOI: 10.1016/j.physd.2016.08.006
      Issue No: Vol. 339 (2016)
       
  • Analysis of the Poisson–Nernst–Planck equation in a ball for modeling
           the Voltage–Current relation in neurobiological microdomains
    • Authors: J. Cartailler; Z. Schuss; D. Holcman
      Pages: 39 - 48
      Abstract: Publication date: 15 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 339
      Author(s): J. Cartailler, Z. Schuss, D. Holcman
      The electro-diffusion of ions is often described by the Poisson–Nernst–Planck (PNP) equations, which couple nonlinearly the charge concentration and the electric potential. This model is used, among others, to describe the motion of ions in neuronal micro-compartments. It remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane. The purpose of this paper is to construct an asymptotic approximation to the solution of the stationary PNP equations in a d -dimensional ball ( d = 1 , 2 , 3 ) in the limit of large total charge. In this geometry the PNP system reduces to the Liouville–Gelfand–Bratú (LGB) equation, with the difference that the boundary condition is Neumann, not Dirichlet, and there is a minus sign in the exponent of the exponential term. The entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson’s equation. These differences replace attraction by repulsion in the LGB equation, thus completely changing the solution. We find that the voltage is maximal in the center and decreases toward the boundary. We also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly, as in classical capacitance theory. This logarithmic singularity is obtained for d = 3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait. These results are used to derive the relation between the outward current and the voltage in a dendritic spine, which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck. This is a fundamental microdomain involved in neuronal communication. We compute the escape rate of an ion from the steady density in a ball, which models a neuronal spine head, to a small absorbing window in the sphere. We predict that the current is defined by the narrow neck that is connected to the sphere by a small absorbing window, as suggested by the narrow escape theory, while voltage is controlled by the PNP equations independently of the neck.

      PubDate: 2016-11-27T16:42:30Z
      DOI: 10.1016/j.physd.2016.09.001
      Issue No: Vol. 339 (2016)
       
  • Numerical analysis of the rescaling method for parabolic problems with
           blow-up in finite time
    • Authors: V.T. Nguyen
      Pages: 49 - 65
      Abstract: Publication date: 15 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 339
      Author(s): V.T. Nguyen
      In this work, we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation u ↦ u λ ( x , t ) : = λ 2 p − 1 u ( λ x , λ 2 t ) . For that purpose, we apply the rescaling method proposed by Berger and Kohn (1988) to such problems. The convergence of the method is proved under some regularity assumption. Some numerical experiments are given to derive the blow-up profile verifying henceforth the theoretical results.

      PubDate: 2016-11-27T16:42:30Z
      DOI: 10.1016/j.physd.2016.09.002
      Issue No: Vol. 339 (2016)
       
  • A Hierarchical Bayes Ensemble Kalman Filter
    • Authors: Michael Tsyrulnikov; Alexander Rakitko
      Pages: 1 - 16
      Abstract: Publication date: 1 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 338
      Author(s): Michael Tsyrulnikov, Alexander Rakitko
      A new ensemble filter that allows for the uncertainty in the prior distribution is proposed and tested. The filter relies on the conditional Gaussian distribution of the state given the model-error and predictability-error covariance matrices. The latter are treated as random matrices and updated in a hierarchical Bayes scheme along with the state. The (hyper)prior distribution of the covariance matrices is assumed to be inverse Wishart. The new Hierarchical Bayes Ensemble Filter (HBEF) assimilates ensemble members as generalized observations and allows ordinary observations to influence the covariances. The actual probability distribution of the ensemble members is allowed to be different from the true one. An approximation that leads to a practicable analysis algorithm is proposed. The new filter is studied in numerical experiments with a doubly stochastic one-variable model of “truth”. The model permits the assessment of the variance of the truth and the true filtering error variance at each time instance. The HBEF is shown to outperform the EnKF and the HEnKF by Myrseth and Omre (2010) in a wide range of filtering regimes in terms of performance of its primary and secondary filters.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.07.009
      Issue No: Vol. 338 (2016)
       
  • The tennis racket effect in a three-dimensional rigid body
    • Authors: Léo Van Damme; Pavao Mardešić; Dominique Sugny
      Pages: 17 - 25
      Abstract: Publication date: 1 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 338
      Author(s): Léo Van Damme, Pavao Mardešić, Dominique Sugny
      We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a three-dimensional rigid body. This effect is characterized by a flip ( π - rotation) of the head of the racket when a full ( 2 π ) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Different examples are discussed.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.07.010
      Issue No: Vol. 338 (2016)
       
  • Stability analysis of amplitude death in delay-coupled high-dimensional
           map networks and their design procedure
    • Authors: Tomohiko Watanabe; Yoshiki Sugitani; Keiji Konishi; Naoyuki Hara
      Pages: 26 - 33
      Abstract: Publication date: 1 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 338
      Author(s): Tomohiko Watanabe, Yoshiki Sugitani, Keiji Konishi, Naoyuki Hara
      The present paper studies amplitude death in high-dimensional maps coupled by time-delay connections. A linear stability analysis provides several sufficient conditions for an amplitude death state to be unstable, i.e., an odd number property and its extended properties. Furthermore, necessary conditions for stability are provided. These conditions, which reduce trial-and-error tasks for design, and the convex direction, which is a popular concept in the field of robust control, allow us to propose a design procedure for system parameters, such as coupling strength, connection delay, and input–output matrices, for a given network topology. These analytical results are confirmed numerically using delayed logistic maps, generalized Henon maps, and piecewise linear maps.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.07.011
      Issue No: Vol. 338 (2016)
       
  • Generalized uncertainty principle and analogue of quantum gravity in
           optics
    • Authors: Maria Chiara Braidotti; Ziad H. Musslimani; Claudio Conti
      Pages: 34 - 41
      Abstract: Publication date: 1 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 338
      Author(s): Maria Chiara Braidotti, Ziad H. Musslimani, Claudio Conti
      The design of optical systems capable of processing and manipulating ultra-short pulses and ultra-focused beams is highly challenging with far reaching fundamental technological applications. One key obstacle routinely encountered while implementing sub-wavelength optical schemes is how to overcome the limitations set by standard Fourier optics. A strategy to overcome these difficulties is to utilize the concept of a generalized uncertainty principle (G-UP) which has been originally developed to study quantum gravity. In this paper we propose to use the concept of G-UP within the framework of optics to show that the generalized Schrödinger equation describing short pulses and ultra-focused beams predicts the existence of a minimal spatial or temporal scale which in turn implies the existence of maximally localized states. Using a Gaussian wavepacket with complex phase, we derive the corresponding generalized uncertainty relation and its maximally localized states. Furthermore, we numerically show that the presence of nonlinearity helps the system to reach its maximal localization. Our results may trigger further theoretical and experimental tests for practical applications and analogues of fundamental physical theories.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.08.001
      Issue No: Vol. 338 (2016)
       
  • Initial–boundary layer associated with the nonlinear
           Darcy–Brinkman–Oberbeck–Boussinesq system
    • Authors: Mingwen Fei; Daozhi Han; Xiaoming Wang
      Pages: 42 - 56
      Abstract: Publication date: 1 January 2017
      Source:Physica D: Nonlinear Phenomena, Volume 338
      Author(s): Mingwen Fei, Daozhi Han, Xiaoming Wang
      In this paper, we study the vanishing Darcy number limit of the nonlinear Darcy–Brinkman–Oberbeck–Boussinesq system (DBOB). This singular perturbation problem involves singular structures both in time and in space giving rise to initial layers, boundary layers and initial–boundary layers. We construct an approximate solution to the DBOB system by the method of multiple scale expansions. The convergence with optimal convergence rates in certain Sobolev norms is established rigorously via the energy method.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.08.002
      Issue No: Vol. 338 (2016)
       
  • Two-dimensional localized structures in harmonically forced oscillatory
           systems
    • Authors: Y.-P. Ma; E. Knobloch
      Pages: 1 - 17
      Abstract: Publication date: 15 December 2016
      Source:Physica D: Nonlinear Phenomena, Volume 337
      Author(s): Y.-P. Ma, E. Knobloch
      Two-dimensional spatially localized structures in the complex Ginzburg–Landau equation with 1:1 resonance are studied near the simultaneous presence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turing-stable and Turing-unstable regimes. In particular, localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defect-mediated snaking in one spatial dimension. Stability of axisymmetric solutions on these branches with respect to axisymmetric and nonaxisymmetric perturbations is determined, and parameter regimes with stable axisymmetric oscillons are identified. Direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction, and of circular and planar localized hexagonal patterns in the fully two-dimensional system.

      PubDate: 2016-10-30T22:02:28Z
      DOI: 10.1016/j.physd.2016.07.003
      Issue No: Vol. 337 (2016)
       
  • A comparison of macroscopic models describing the collective response of
           sedimenting rod-like particles in shear flows
    • Authors: Christiane Helzel; Athanasios E. Tzavaras
      Pages: 18 - 29
      Abstract: Publication date: 15 December 2016
      Source:Physica D: Nonlinear Phenomena, Volume 337
      Author(s): Christiane Helzel, Athanasios E. Tzavaras
      We consider a kinetic model, which describes the sedimentation of rod-like particles in dilute suspensions under the influence of gravity, presented in Helzel and Tzavaras (submitted for publication). Here we restrict our considerations to shear flow and consider a simplified situation, where the particle orientation is restricted to the plane spanned by the direction of shear and the direction of gravity. For this simplified kinetic model we carry out a linear stability analysis and we derive two different nonlinear macroscopic models which describe the formation of clusters of higher particle density. One of these macroscopic models is based on a diffusive scaling, the other one is based on a so-called quasi-dynamic approximation. Numerical computations, which compare the predictions of the macroscopic models with the kinetic model, complete our presentation.

      PubDate: 2016-10-30T22:02:28Z
      DOI: 10.1016/j.physd.2016.07.004
      Issue No: Vol. 337 (2016)
       
  • Exploiting stiffness nonlinearities to improve flow energy capture from
           the wake of a bluff body
    • Authors: Ali H. Alhadidi; Hamid Abderrahmane; Mohammed F. Daqaq
      Pages: 30 - 42
      Abstract: Publication date: 15 December 2016
      Source:Physica D: Nonlinear Phenomena, Volume 337
      Author(s): Ali H. Alhadidi, Hamid Abderrahmane, Mohammed F. Daqaq
      Fluid–structure coupling mechanisms such as wake galloping have been recently utilized to develop scalable flow energy harvesters. Unlike traditional rotary-type generators which are known to suffer serious scalability issues because their efficiency drops significantly as their size decreases; wake-galloping flow energy harvesters (FEHs) operate using a very simple motion mechanism, and, hence can be scaled down to fit the desired application. Nevertheless, wake-galloping FEHs have their own shortcomings. Typically, a wake-galloping FEH has a linear restoring force which results in a very narrow lock-in region. As a result, it does not perform well under the broad range of shedding frequencies normally associated with a variable flow speed. To overcome this critical problem, this article demonstrates theoretically and experimentally that, a bi-stable restoring force can be used to broaden the steady-state bandwidth of wake galloping FEHs and, thereby to decrease their sensitivity to variations in the flow speed. An experimental case study is carried out in a wind tunnel to compare the performance of a bi-stable and a linear FEH under single- and multi-frequency vortex street. An experimentally-validated lumped-parameters model of the bi-stable harvester is also introduced, and solved using the method of multiple scales to study the influence of the shape of the potential energy function on the output voltage.

      PubDate: 2016-10-30T22:02:28Z
      DOI: 10.1016/j.physd.2016.07.005
      Issue No: Vol. 337 (2016)
       
  • Variety of strange pseudohyperbolic attractors in three-dimensional
           generalized Hénon maps
    • Authors: A.S. Gonchenko; S.V. Gonchenko
      Pages: 43 - 57
      Abstract: Publication date: 15 December 2016
      Source:Physica D: Nonlinear Phenomena, Volume 337
      Author(s): A.S. Gonchenko, S.V. Gonchenko
      In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has positive maximal Lyapunov exponent and this property is robust, i.e., it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized Hénon maps of form x ̄ = y , y ̄ = z , z ̄ = B x + A z + C y + g ( y , z ) , where A , B and C are parameters ( B is the Jacobian) and g ( 0 , 0 ) = g ′ ( 0 , 0 ) = 0 .

      PubDate: 2016-10-30T22:02:28Z
      DOI: 10.1016/j.physd.2016.07.006
      Issue No: Vol. 337 (2016)
       
  • Oscillatory instabilities of gap solitons in a repulsive
           Bose–Einstein condensate
    • Authors: P.P. Kizin; D.A. Zezyulin; G.L. Alfimov
      Pages: 58 - 66
      Abstract: Publication date: 15 December 2016
      Source:Physica D: Nonlinear Phenomena, Volume 337
      Author(s): P.P. Kizin, D.A. Zezyulin, G.L. Alfimov
      The paper is devoted to numerical study of stability of nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross–Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest (“fundamental”) gap solitons in the first and in the second spectral gap undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The number of unstable eigenvalues and the associated instability rates are described. Several stable and unstable more complex (non-fundamental) gap solitons are also discussed. The results obtained from the Evans function approach are independently confirmed using the direct numerical integration of the GPE.

      PubDate: 2016-10-30T22:02:28Z
      DOI: 10.1016/j.physd.2016.07.007
      Issue No: Vol. 337 (2016)
       
  • Limit cycles in planar piecewise linear differential systems with
           nonregular separation line
    • Authors: Pedro Toniol Cardin; Joan Torregrosa
      Pages: 67 - 82
      Abstract: Publication date: 15 December 2016
      Source:Physica D: Nonlinear Phenomena, Volume 337
      Author(s): Pedro Toniol Cardin, Joan Torregrosa
      In this paper we deal with planar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles α and 2 π − α , respectively, for α ∈ ( 0 , π ) . We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of α , we prove the existence of systems with four limit cycles up to fifth order and, for α = π / 2 , we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line.

      PubDate: 2016-10-30T22:02:28Z
      DOI: 10.1016/j.physd.2016.07.008
      Issue No: Vol. 337 (2016)
       
  • On Wright’s generalized Bessel kernel
    • Authors: Lun Zhang
      Pages: 92 - 119
      Abstract: Publication date: 1 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 340
      Author(s): Lun Zhang
      In this paper, we consider the Wright’s generalized Bessel kernel K ( α , θ ) ( x , y ) defined by θ x α ∫ 0 1 J α + 1 θ , 1 θ ( u x ) J α + 1 , θ ( ( u y ) θ ) u α d u , α > − 1 , θ > 0 , where J a , b ( x ) = ∑ j = 0 ∞ ( − x ) j j ! Γ ( a + b j ) , a ∈ C , b > − 1 , is Wright’s generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to θ = 1 ) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib–Borodin ensembles. We show that, if θ is rational, i.e., θ = m n with m , n ∈ N , g c d ( m , n ) = 1 , and α > m − 1 − m n , the Wright’s generalized Bessel kernel is integrable in the sense of Its–Izergin–Korepin–Slavnov. We then come to the Fredholm determinant of this kernel over the union of several scaled intervals, which can also be interpreted as the gap probability (the probability of finding no particles) on these intervals. The integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding Fredholm determinant as well as a Hamiltonian interpretation. As a consequence, we are able to represent the gap probability over a single interval ( 0 , s ) in terms of a solution of a system of nonlinear ordinary differential equations.

      PubDate: 2016-12-18T03:09:08Z
      DOI: 10.1016/j.jat.2016.09.002
      Issue No: Vol. 213 (2016)
       
  • Windows of opportunity for synchronization in stochastically coupled maps
    • Authors: Olga Golovneva; Russell Jeter Igor Belykh Maurizio Porfiri
      Abstract: Publication date: 1 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 340
      Author(s): Olga Golovneva, Russell Jeter, Igor Belykh, Maurizio Porfiri
      Several complex systems across science and engineering display on–off intermittent coupling among their units. Most of the current understanding of synchronization in switching networks relies on the fast switching hypothesis, where the network dynamics evolves at a much faster time scale than the individual units. Recent numerical evidence has demonstrated the existence of windows of opportunity, where synchronization may be induced through non-fast switching. Here, we study synchronization of coupled maps whose coupling gains stochastically switch with an arbitrary switching period. We determine the role of the switching period on synchronization through a detailed analytical treatment of the Lyapunov exponent of the stochastic dynamics. Through closed-form expressions and numerical findings, we demonstrate the emergence of windows of opportunity and elucidate their nontrivial relationship with the stability of synchronization under static coupling. Our results are expected to provide a rigorous basis for understanding the dynamic mechanisms underlying the emergence of windows of opportunity and leverage non-fast switching in the design of evolving networks.
      Graphical abstract image

      PubDate: 2016-12-18T03:09:08Z
       
  • Arnold tongues in a billiard problem in nonlinear and nonequilibrium
           systems
    • Authors: Tomoyuki Miyaji
      Abstract: Publication date: 1 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 340
      Author(s): Tomoyuki Miyaji
      We study a billiard problem in nonlinear and nonequilibrium systems. This is motivated by the motions of a traveling spot in a reaction–diffusion system (RDS) in a rectangular domain. We consider a four-dimensional dynamical system, defined by ordinary differential equations. This was first derived by S.-I. Ei et al. (2006), based on a reduced system on the center manifold in a neighborhood of a pitchfork bifurcation of a stationary spot for the RDS. In contrast to the classical billiard problem, this defines a dynamical system that is dissipative rather than conservative, and has an attractor. According to previous numerical studies, the attractor of the system changes depending on parameters such as the aspect ratio of the domain. It may be periodic, quasi-periodic, or chaotic. In this paper, we elucidate that it results from parameters crossing Arnold tongues and that the organizing center is a Hopf–Hopf bifurcation of the trivial equilibrium.

      PubDate: 2016-12-18T03:09:08Z
       
  • Corrigendum to “Dynamics and stability of a discrete breather in a
           harmonically excited chain with vibro-impact on-site potential” [Physica
           D 292–293 (2015) 8–28]
    • Authors: Nathan Perchikov; O.V. Gendelman
      Abstract: Publication date: 1 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 340
      Author(s): Nathan Perchikov, O.V. Gendelman


      PubDate: 2016-12-18T03:09:08Z
       
  • Timing variation in an analytically solvable chaotic system
    • Authors: J.N. Blakely; M.S. Milosavljevic N.J. Corron
      Abstract: Publication date: 1 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 340
      Author(s): J.N. Blakely, M.S. Milosavljevic, N.J. Corron
      We present analytic solutions for a chaotic dynamical system that do not have the regular timing characteristic of recently reported solvable chaotic systems. The dynamical system can be viewed as a first order filter with binary feedback. The feedback state may be switched only at instants defined by an external clock signal. Generalizing from a period one clock, we show analytic solutions for period two and higher period clocks. We show that even when the clock ‘ticks’ randomly the chaotic system has an analytic solution. These solutions can be visualized in a stroboscopic map whose complexity increases with the complexity of the clock. We provide both analytic results as well as experimental data from an electronic circuit implementation of the system. Our findings bridge the gap between the irregular timing of well known chaotic systems such as Lorenz and Rossler and the well regulated oscillations of recently reported solvable chaotic systems.
      Graphical abstract image

      PubDate: 2016-12-18T03:09:08Z
       
  • Data-based stochastic model reduction for the Kuramoto–Sivashinsky
           equation
    • Authors: Fei Kevin; Lin Alexandre Chorin
      Abstract: Publication date: 1 February 2017
      Source:Physica D: Nonlinear Phenomena, Volume 340
      Author(s): Fei Lu, Kevin K. Lin, Alexandre J. Chorin
      The problem of constructing data-based, predictive, reduced models for the Kuramoto–Sivashinsky equation is considered, under circumstances where one has observation data only for a small subset of the dynamical variables. Accurate prediction is achieved by developing a discrete-time stochastic reduced system, based on a NARMAX (Nonlinear Autoregressive Moving Average with eXogenous input) representation. The practical issue, with the NARMAX representation as with any other, is to identify an efficient structure, i.e., one with a small number of terms and coefficients. This is accomplished here by estimating coefficients for an approximate inertial form. The broader significance of the results is discussed.

      PubDate: 2016-12-18T03:09:08Z
       
  • Numerical bifurcation for the capillary Whitham equation
    • Authors: Filippo Remonato; Henrik Kalisch
      Abstract: Publication date: Available online 6 December 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Filippo Remonato, Henrik Kalisch
      The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth. In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Whitham equation is a nonlocal equation similar to the usual Whitham equation, but containing an additional term with a coefficient depending on the Bond number which measures the relative strength of capillary and gravity effects on the wave motion. A spectral collocation scheme for computing approximations to periodic traveling waves for the capillary Whitham equation is put forward. Numerical approximations of periodic traveling waves are computed using a bifurcation approach, and a number of bifurcation curves are found. Our analysis uncovers a rich structure of bifurcation patterns, including subharmonic bifurcations, as well as connecting and crossing branches. Indeed, for some values of the Bond number, the bifurcation diagram features distinct branches of solutions which intersect at a secondary bifurcation point. The same branches may also cross without connecting, and some bifurcation curves feature self-crossings without self-connections.

      PubDate: 2016-12-11T02:34:16Z
      DOI: 10.1016/j.physd.2016.11.003
       
  • One- and two-dimensional bright solitons in inhomogeneous defocusing
           nonlinearities with an antisymmetric periodic gain and loss
    • Authors: Dengchu Guo; Jing Xiao; Linlin Gu; Hongzhen Jin; Liangwei Dong
      Abstract: Publication date: Available online 2 December 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dengchu Guo, Jing Xiao, Linlin Gu, Hongzhen Jin, Liangwei Dong
      We address that various branches of bright solitons exist in a spatially inhomogeneous defocusing nonlinearity with an imprinted antisymmetric periodic gain-loss profile. The spectra of such systems with a purely imaginary potential never become complex and thus the parity-time symmetry is unbreakable. The mergence between pairs of soliton branches occurs at a critical gain-loss strength, above which no soliton solutions can be found. Intriguingly, which pair of soliton branches will merge together can be changed by varying the modulation frequency of gain and loss. Most branches of one-dimensional solitons are stable in wide parameter regions. We also provide the first example of two-dimensional bright solitons with unbreakable parity-time symmetry.

      PubDate: 2016-12-04T02:03:09Z
      DOI: 10.1016/j.physd.2016.11.005
       
  • Excitability, mixed-mode oscillations and transition to chaos in a
           stochastic ice ages model
    • Authors: D.V. Alexandrov; I.A. Bashkirtseva; L.B. Ryashko
      Abstract: Publication date: Available online 30 November 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): D.V. Alexandrov, I.A. Bashkirtseva, L.B. Ryashko
      Motivated by an important geophysical significance, we consider the influence of stochastic forcing on a simple three-dimensional climate model previously derived by Saltzman and Sutera. A nonlinear dynamical system governing three physical variables, the bulk ocean temperature, continental and marine ice masses, is analyzed in deterministic and stochastic cases. It is shown that the attractor of deterministic model is either a stable equilibrium or a limit cycle. We demonstrate that the process of continental ice melting occurs with a noise-dependent time delay as compared with marine ice melting. The paleoclimate cyclicity which is near 100 ky in a wide range of model parameters abruptly increases in the vicinity of a bifurcation point and depends on the noise intensity. In a zone of stable equilibria, the 3D climate model under consideration is extremely excitable. Even for a weak random noise, the stochastic trajectories demonstrate a transition from small- to large-amplitude stochastic oscillations (SLASO). In a zone of stable cycles, SLASO transitions are analyzed too. We show that such stochastic transitions play an important role in the formation of a mixed-mode paleoclimate scenario. This mixed-mode dynamics with the intermittency of large- and small-amplitude stochastic oscillations and coherence resonance are investigated via analysis of interspike intervals. A tendency of dynamic paleoclimate to abrupt and rapid glaciations and deglaciations as well as its transition from order to chaos with increasing noise are shown.

      PubDate: 2016-12-04T02:03:09Z
      DOI: 10.1016/j.physd.2016.11.007
       
  • Optical dispersive shock waves in defocusing colloidal media
    • Authors: X. An; T.R. Marchant; N.F. Smyth
      Abstract: Publication date: Available online 24 November 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): X. An, T.R. Marchant, N.F. Smyth
      The propagation of an optical dispersive shock wave, generated from a jump discontinuity in light intensity, in a defocussing colloidal medium is analysed. The equations governing nonlinear light propagation in a colloidal medium consist of a nonlinear Schrödinger equation for the beam and an algebraic equation for the medium response. In the limit of low light intensity, these equations reduce to a perturbed higher order nonlinear Schrödinger equation. Solutions for the leading and trailing edges of the colloidal dispersive shock wave are found using modulation theory. This is done for both the perturbed nonlinear Schrödinger equation and the full colloid equations for arbitrary light intensity. These results are compared with numerical solutions of the colloid equations.

      PubDate: 2016-11-27T16:42:30Z
      DOI: 10.1016/j.physd.2016.11.004
       
  • Spatiotemporal control to eliminate cardiac alternans using isostable
           reduction
    • Authors: Dan Wilson; Jeff Moehlis
      Abstract: Publication date: Available online 16 November 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dan Wilson, Jeff Moehlis
      Cardiac alternans, an arrhythmia characterized by a beat-to-beat alternation of cardiac action potential durations, is widely believed to facilitate the transition from normal cardiac function to ventricular fibrillation and sudden cardiac death. Alternans arises due to an instability of a healthy period-1 rhythm, and most dynamical control strategies either require extensive knowledge of the cardiac system, making experimental validation difficult, or are model independent and sacrifice important information about the specific system under study. Isostable reduction provides an alternative approach, in which the response of a system to external perturbations can be used to reduce the complexity of a cardiac system, making it easier to work with from an analytical perspective while retaining many of its important features. Here, we use isostable reduction strategies to reduce the complexity of partial differential equation models of cardiac systems in order to develop energy optimal strategies for the elimination of alternans. Resulting control strategies require significantly less energy to terminate alternans than comparable strategies and do not require continuous state feedback.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.11.001
       
  • Macroscopic heat transport equations and heat waves in nonequilibrium
           states
    • Authors: Yangyu Guo; David Jou; Moran Wang
      Abstract: Publication date: Available online 16 November 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yangyu Guo, David Jou, Moran Wang
      Heat transport may behave as wave propagation when the time scale of processes decreases to be comparable to or smaller than the relaxation time of heat carriers. In this work, a generalized heat transport equation including nonlinear, nonlocal and relaxation terms is proposed, which sums up the Cattaneo-Vernotte, dual-phase-lag and phonon hydrodynamic models as special cases. In the frame of this equation, the heat wave propagations are investigated systematically in nonequilibrium steady states, which were usually studied around equilibrium states. The phase (or front) speed of heat waves is obtained through a perturbation solution to the heat differential equation, and found to be intimately related to the nonlinear and nonlocal terms. Thus, potential heat wave experiments in nonequilibrium states are devised to measure the coefficients in the generalized equation, which may throw light on understanding the physical mechanisms and macroscopic modeling of nanoscale heat transport.

      PubDate: 2016-11-20T14:57:33Z
      DOI: 10.1016/j.physd.2016.10.005
       
 
 
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